A heuristic for ray class groups of quadratic number fields
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908556774604800 |
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| author | Bartel, Alex Pagano, Carlo |
| author_facet | Bartel, Alex Pagano, Carlo |
| contents | We formulate a model for the average behaviour of ray class groups of real quadratic fields with respect to a fixed rational modulus, locally at a finite set $S$ of odd primes. To that end, we introduce Arakelov ray class groups of a number field, and postulate that, locally at $S$, the Arakelov ray class groups of real quadratic fields are distributed randomly with respect to a natural Cohen--Lenstra type probability measure. We show that our heuristics imply the Cohen--Lenstra heuristics on class groups of real quadratic fields, as well as equidistribution results on the fundamental unit of a real quadratic field modulo an integer, and are consistent with Varma's results on average of sizes of $3$-torsion subgroups of ray class groups of quardratic fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_20185 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A heuristic for ray class groups of quadratic number fields Bartel, Alex Pagano, Carlo Number Theory 11R65, 11R11, 11R29 We formulate a model for the average behaviour of ray class groups of real quadratic fields with respect to a fixed rational modulus, locally at a finite set $S$ of odd primes. To that end, we introduce Arakelov ray class groups of a number field, and postulate that, locally at $S$, the Arakelov ray class groups of real quadratic fields are distributed randomly with respect to a natural Cohen--Lenstra type probability measure. We show that our heuristics imply the Cohen--Lenstra heuristics on class groups of real quadratic fields, as well as equidistribution results on the fundamental unit of a real quadratic field modulo an integer, and are consistent with Varma's results on average of sizes of $3$-torsion subgroups of ray class groups of quardratic fields. |
| title | A heuristic for ray class groups of quadratic number fields |
| topic | Number Theory 11R65, 11R11, 11R29 |
| url | https://arxiv.org/abs/2509.20185 |